arX

iv:1

005.

4678

v3 [

hep-

ph]

28

Sep

2010

UCI-TR-2010-06

Sommerfeld Enhancements for Thermal Relic Dark Matter

Jonathan L. Feng, Manoj Kaplinghat, and Hai-Bo Yu

Department of Physics and Astronomy,

University of California, Irvine, California 92697, USA

(Dated: August 2010)

Abstract

The annihilation cross section of thermal relic dark matter determines both its relic density

and indirect detection signals. We determine how large indirect signals may be in scenarios with

Sommerfeld-enhanced annihilation, subject to the constraint that the dark matter has the correct

relic density. This work refines our previous analysis through detailed treatments of resonant

Sommerfeld enhancement and the effect of Sommerfeld enhancement on freeze out. Sommerfeld

enhancements raise many interesting issues in the freeze out calculation, and we find that the

cutoff of resonant enhancement, the equilibration of force carriers, the temperature of kinetic

decoupling, and the efficiency of self-interactions for preserving thermal velocity distributions all

play a role. These effects may have striking consequences; for example, for resonantly-enhanced

Sommerfeld annihilation, dark matter freezes out but may then chemically recouple, implying

highly suppressed indirect signals, in contrast to naive expectations. In the minimal scenario

with standard astrophysical assumptions, and tuning all parameters to maximize the signal, we

find that, for force-carrier mass mφ = 250 MeV and dark matter masses mX = 0.1, 0.3, and 1

TeV, the maximal Sommerfeld enhancement factors are Seff = 7, 30, and 90, respectively. Such

boosts are too small to explain both the PAMELA and Fermi excesses. Non-minimal models may

require smaller boosts, but the bounds on Seff could also be more stringent, and dedicated freeze

out analyses are required. For concreteness, we focus on 4µ final states, but we also discuss 4e

and other modes, deviations from standard astrophysical assumptions and non-minimal particle

physics models, and we outline the steps required to determine if such considerations may lead to

a self-consistent explanation of the PAMELA or Fermi excesses.

PACS numbers: 95.35.+d, 95.85.Ry

1

I. INTRODUCTION

Dark matter may be composed of thermal relics with mass near the weak scale mweak ∼100 GeV− 1 TeV. Such candidates are produced in the hot early Universe and then freezeout when the Universe cools and expands. Their annihilation cross section therefore deter-mines both the relic density and the rate of annihilation today. The requirement that thecandidate be much or all of the dark matter therefore constrains its annihilation rate now,with important implications for indirect searches for dark matter.

Although annihilation in the early Universe and now is determined by the same dynamics,the kinematics are vastly different: at freeze out, thermal relics have relative velocity vrel ∼0.3, whereas today, vrel ∼ 10−3. As a result, the numerical values of the annihilation crosssections may differ significantly. For example, neutralino annihilation is in many casesdominated by P -wave processes [1], and so is suppressed at low velocities. In this work, weconsider Sommerfeld-enhanced cross sections, which have the opposite behavior: they areenhanced at low velocities and therefore boost present-day annihilation signals. We refinea previous study [2] and examine how large these boosts may be, subject to the constraintthat the thermal relic has the correct thermal relic density to be dark matter.

Along with its general implications for future indirect searches, this study also has directimplications for the interpretation of current data. Following earlier excesses reported by theHEAT Collaboration [3, 4], the PAMELA [5], ATIC [6], and Fermi [7] Collaborations havereported excesses of cosmic positrons over an estimate of expected background [8]. Darkmatter annihilation with Sommerfeld enhancement [9] of dark matter annihilation [10], gen-eralized to massive force carriers [11–15], has been proposed as an explanation [16–19]. Inits simplest form, this scenario assumes that dark matter is composed of a single particlespecies X , which interacts with light force carriers φ with fine structure constant αX andmφ ≪ mX ∼ mweak. This new interaction modifies dark matter annihilation and scatter-ing properties. The resulting annihilation cross section multiplied by relative velocity is(σanvrel)0S, where (σanvrel)0 is its tree-level value, and S is Sommerfeld’s original enhance-ment factor [9]

S0 =π αX/v

1− e−παX/v

αX≫v−→ παX

v, (1)

generalized to massive φ, where v ≡ vrel/2 is the dark matter particle’s velocity in thecenter-of-mass frame. In the proposed explanation, dark matter freezes out at early timeswhen the Sommerfeld effect is negligible with (σanvrel)0 ≈ 3 × 10−26 cm3/s, leading to thecorrect relic density ΩDMh

2 ≃ 0.114. At present, however, when vrel is much smaller, theSommerfeld effect becomes important, and, for example, for mX ∼ 2 TeV and an assumedenhancement factor of S ∼ 1000, such annihilations are sufficient to explain the positronexcesses [20, 21].

In Ref. [2], we showed that in straightforward models, such large Sommerfeld enhance-ments cannot be self-consistently realized. The problem is simple to state: large Sommerfeldenhancement requires strong interactions, and strongly-interacting particles annihilate tooefficiently in the early Universe to be all of the dark matter. More quantitatively, the re-quired new force carrier interaction necessarily induces an annihilation process XX → φφ,with cross section

(σanvrel)0 ∼πα2

X

m2X

. (2)

Conservatively neglecting other annihilation processes, for a typical weak-scale mass mX ,

2

the thermal relic density implies a typical weak coupling αX ≈ 0.05 [mX/2 TeV], as dic-tated by the WIMP miracle. The resulting Sommerfeld enhancement now is ∼ παX/v ≈100 [mX/2 TeV], an order of magnitude too small to explain the PAMELA and Fermi signals.Alternatively, to achieve S ∼ 1000, one requires αX ∼ 1, which implies ΩX ∼ 0.001, twoorders of magnitude smaller than the value assumed in deriving the requirement S ∼ 1000.

In this work, we refine our previous analysis in several ways. In Ref. [2] we approximatedthe Sommerfeld enhancement S by its value at mφ = 0, given by S0 in Eq. (1). Formassive φ, the Sommerfeld enhancement cuts off at a value proportional to mX/mφ andalso exhibits resonant structure. Here we use numerical results (and a highly accurateanalytic approximation to the numerical results [22–25]) for S.

In addition, we refine our previous work to include the effect of Sommerfeld enhancementon freeze out. One might expect this effect to be negligible, because freeze out is typicallythought to occur at v ∼ 0.3, when the Sommerfeld effect is insignificant. However, anni-hilation continues to much later times, and Sommerfeld enhancement has an impact whenthe dark matter cools. Sommerfeld effects on freeze out have been considered previously inRef. [26] for the case of Wino dark matter and in Refs. [27–30] for hidden sector dark mat-ter. In the analyses published after the PAMELA and Fermi excesses were reported [28–30],(σanvrel)0 was taken as a free parameter, and the fact that it depends on the same parametersthat determine S was not used. Here we make essential use of the observation that there isa irreducible contribution to (σanvrel)0 of the form of Eq. (2), and so constraints on the relicdensity bound the fundamental parameters that also determine S.

As we will see, the combination of these two refinements highlights a number of effectsthat may typically be ignored, but now require exploration. For example, the relic density isin principle affected by the cutoff of resonant Sommerfeld enhancement, the production anddecay rates of force carrier particles, the temperature of kinetic decoupling, and the efficiencyof self-scattering for preserving the dark matter’s thermal velocity distribution. Very closeto resonances, we also find the intriguing possibility of chemical recoupling, in which darkmatter freezes out and then melts back in, with annihilations becoming important againat late times. This implies that exact resonances suppress, rather than enhance, indirectsignals.

We examine the quantitative impact of all of these effects, and present results for themaximal Sommerfeld enhancement of indirect signals. In the most optimistic case, thatis, tuning all parameters to maximize the signal, we find that, for mφ = 250 MeV andmX = 0.1, 0.3, and 1 TeV, the largest possible effective Sommerfeld enhancements areSeff = 7, 30, and 90, respectively. This refined analysis therefore strengthens our previousresults: Sommerfeld enhancements may be significant, but they fall short of explaining thecurrent PAMELA and Fermi cosmic ray excesses. We then discuss various astrophysicaleffects that may reduce the discrepancy between the maximal enhancements derived hereand those required to explain the data, and critically examine several non-minimal modelsand the issues that must be addressed to determine if more complicated particle physicsmodels may provide viable explanations.

This paper is organized as follows. In Sec. II, we describe our underlying model as-sumptions and our treatment of Sommerfeld enhancements with resonances. In Sec. III weanalyze the effect of Sommerfeld enhancement on freeze out. In Sec. IV we assemble thesepieces and present the results for the maximal Sommerfeld enhancement achievable now. InSec. V, we compare these to the current data, and discuss non-standard astrophysical effectsand non-minimal particle physics models. We present our conclusions in Sec. VI.

3

II. SOMMERFELD ENHANCEMENT WITH RESONANCES

We consider a simple model with a hidden sector dark matter particle X , which cou-ples to a light force carrier φ with coupling

√4παX . The annihilation cross section is then

(σanvrel)0S, where (σanvrel)0 is the tree-level cross section and S is the Sommerfeld enhance-ment.

To maximize the Sommerfeld enhancement, we assume that (σanvrel)0 is dominated byS-wave processes, and so is unsuppressed at low velocities. We also consider only the“irreducible” annihilation channel XX → φφ, and take the tree-level cross section

(σanvrel)0 =πα2

X

m2X

. (3)

This may be modified by O(1) pre-factors, depending, for example, on whether X is aMajorana or Dirac fermion, and whether φ is a scalar or a gauge boson. In addition,even in simple models, (σanvrel)0 will typically receive additional contributions from otherannihilation channels; to maximize the Sommerfeld effect on indirect search signals, weneglect these other channels here, but discuss their impact in Sec. V.

To determine the enhancement factor S, we numerically solve the differential equation

1

mX

d2χ

dr2+

αX

re−mφrχ = −mXv

2χ , (4)

with the boundary conditions χ′(r) = imXvχ(r) and χ(r) = eimXvr when r → ∞. TheSommerfeld enhancement factor is given by

S =|χ(∞)|2

|χ(0)|2. (5)

The Sommerfeld enhancement may also be obtained by approximating the Yukawa po-tential by the Hulthen potential, for which an analytic solution is possible [24]. The resultinganalytic approximation to the Sommerfeld enhancement is [24, 25]

S =π

ǫv

sinh(

2πǫvπ2ǫφ/6

)

cosh(

2πǫvπ2ǫφ/6

)

− cos(

2π√

1π2ǫφ/6

− ǫ2v(π2ǫφ/6)2

) , (6)

where ǫv ≡ v/αX and ǫφ ≡ mφ/(αXmX). The analytic expression of Eq. (6) is comparedto the numerical solution in Fig. 1. We see that the analytic result is an excellent approxi-mation, typically reproducing the numerical results to within fractional differences of 10%,and accurately reproducing the resonant behavior. Given Eq. (6), we see that for ǫv ≪ ǫφ,these resonances are at

mφ ≃ 6αXmX

π2n2, n = 1, 2, 3 . . . (7)

At these resonances, with mφ determined by Eq. (7), the Sommerfeld enhancement for lowv is

S ≃ π2αXmφ

6mXv2. (8)

S is therefore enhanced by v−2 at resonances, as opposed to v−1 away from resonances.

4

10-2 10-1 100100

101

102

103

104

105

106

107

Sv=10

-3

10-2 10-1 100100

101

102

103

104

105

106

107

108

109

S

v

FIG. 1: The Sommerfeld enhancement factor S as a function of ǫφ ≡ mφ/(αXmX) for the constant

values of ǫv ≡ v/αX indicated. The solid red curves are the analytic approximation of Eq. (6), and

the dashed blue curves are numerical results.

Note, however, that both the numerical and analytic results become infinite on resonance.This is unphysical, a result of the fact that the quantum mechanical treatments do notinclude the effect of bound state decay. In fact, the finite lifetime of bound states impliesthat S saturates at v ∼ α3

X(mφ/mX) [13], which, given the condition for a strong resonance,Eq. (7) with small n, is v ∼ α4

X . In this study, we use the analytic form for S given inEq. (6) and include the saturation by making the substitution v → v + α4

X .

III. THERMAL FREEZE OUT WITH SOMMERFELD ENHANCEMENT

A. General Formalism

Given values of mX , mφ, and αX , we must determine the relic density ΩX . The formalismof thermal freeze out is well developed [31, 32]. Here we review this formalism in sufficientgenerality to accommodate novel effects resulting from Sommerfeld enhancements.

The evolution of the abundance of a thermal relic X is governed by the Boltzmannequation

dnX

dt+ 3HnX = −〈σanvrel〉

(

n2X − neq 2

X

)

, (9)

where nX is the number density of the dark matter particles, neqX is its value in equilibrium,

H is the Hubble constant, and 〈σanvrel〉 is the annihilation cross section multiplied by therelative velocity, averaged over the dark matter velocity distribution. Changing variablesfrom t → x = mX/T and nX → Y = nX/s, where s is the entropy density, the Boltzmannequation becomes

dY

dx= −

√

π

45mPlmX

(g∗s/√g∗ )

x2〈σanvrel〉

(

Y 2 − Y eq 2)

, (10)

where mPl = G−1/2N ≃ 1.2 × 1019 GeV is the Planck mass, and g∗s and g∗ are the effective

relativistic degrees of freedom for entropy and energy density, respectively. In this section

5

we will assume that the particles X annihilates to are in thermal equilibrium at the timeof freeze out. We will revisit this issue in Sec. III B, where we show that this requirementleads to significant constraints if the dominant annihilation is to the dark force carriers.

To evaluate the relic density, we first evolve it to the time of chemical decoupling, whenthe annihilation rate Γan ≡ nX〈σanvrel〉 is approximately the expansion rate H . As discussedin Sec. II, we assume the annihilation cross section has the form σanvrel = (σanvrel)0S, where(σanvrel)0 is the S-wave, tree-level cross section, and S is the Sommerfeld enhancement. Upto the time of freeze out, the Sommerfeld effect is insignificant, and so the standard resultsfor S-wave annihilators apply: defining freeze out by

Y (xf ) = (1 + c) Y eq(xf) , (11)

where c is a constant, freeze out occurs at

xf ≈ ln ξ − 1

2ln (ln ξ)

ξ = 0.038 c (2 + c)mPlmX (g/√g∗ ) (σanvrel)0 , (12)

where g is the number of degrees of freedom of the dark matter particle; we take g = 2. Tomatch numerical results, c ∼ 1; we choose c = 1 and have checked that varying c from 1/2to 2 has no appreciable effect. To further test this approximation we have also numericallyevolved Eq. (10) from x = 20 to x = 100 (when the equilibrium abundance is effectivelyzero) and then used Eq. (12) to evolve into the late Universe. This numerical calculationtypically results in 10% smaller relic abundance. Given the excellent agreement, we set c = 1and with this choice, Eqs. (11) and (12) determine Y (xf ), the abundance at freeze out.

After freeze out, Y eq quickly becomes insignificant. Neglecting it, we may solve Eq. (10)to find

1

Y (xs)=

1

Y (xf)+

√

π

45mPlmX

∫ xkd

xf

(g∗s/√g∗)〈σanvrel〉x2

dx

+

√

π

45mPl mX

∫ xs

xkd

(g∗s/√g∗)〈σanvrel〉x2

dx , (13)

where xkd is the value of x at kinetic decoupling and xs is its value when annihilationsbecome insignificant and we may stop the integration. We have broken the integral into twoto emphasize that there are two eras: before and after the temperature of kinetic decouplingTkd. Before Tkd, the dark matter’s velocity distribution is thermal with temperature TX = T .After Tkd, the dark matter’s velocity distribution initially remains thermal, but TX drops asa−2, while T drops as a−1, where a is the scale factor, and so TX = T 2/Tkd. Eventually, thedark matter’s velocity distribution need not even be thermal. We discuss the value of Tkd

and the issue of non-thermal velocity distributions in Secs. III C and IIID, respectively.When the dark matter distribution is thermal with temperature TX , the thermally-

averaged cross section in the non-relativistic limit is

〈σanvrel〉 ≡∫

f(~v1)f(~v2)σanvreld3~v1d

3~v2

=∫

√

2

π

1

v30v2rele

−v2rel

2v20 σanvreldvrel , (14)

6

where f(~v) is the dark matter’s velocity distribution, vrel = |~v1 − ~v2|, and

v0 =

√

2TX

mX

≡√

2

xX

, (15)

the most probable velocity. For the Sommerfeld-enhanced annihilation cross section, thisbecomes

〈σanvrel〉 =x3/2X

2√π

∫ ∞

0(σanvrel)v

2rele

−xXv2rel

/4dvrel = (σanvrel)0S(xX) , (16)

where

S(xX) =x3/2X

2√π

∫

∞

0Sv2rele

−xXv2rel

/4dvrel (17)

is the Sommerfeld enhancement averaged over a thermal distribution with temperature TX =mX/xX .

B. Equilibration of Force Carriers

In the relic density calculation, we have implicitly assumed that, when the X particleschemically decouple, the force carriers φ are in thermal equilibrium with the massless par-ticles of the standard model. The force carriers are expected to interact with the standardmodel, as their decays to standard model particles typically provide the indirect signals. Ina simple example, if φ is a U(1) gauge boson, it may mix with the standard model photonthrough kinetic mixing terms ∼ ǫFEM

µν F φµν . After diagonalizing the φ-photon system, stan-dard model particles with charge Q have hidden charge ǫQ [33], and so the φ particles decaythrough φ → f f , where f = e, µ, u, d, s, . . .. The largest kinetic mixing parameter allowedby current particle physics constraints is ǫ ∼ 10−3 [34, 35].

The existence of φ interactions with the visible sector does not, however, guarantee thatthey are efficient enough to bring the φ particles in thermal equilibrium. Here we determinesufficient conditions for the kinetic mixing example to guarantee the equilibration of φparticles. The leading φ number-changing interactions between the φ particles and the visiblesector are decays φ → f f and inverse decays f f → φ. Other φ number-changing processes,such as f f → φγ, qq → φg, and fγ → fφ, are parametrically suppressed compared tothese and subdominant. For temperatures T ≫ mφ, the decay and inverse decay processesbalance (since f, f are in chemical equilibrium), and we may simply compare the decay rateto the expansion rate [36].

For T ≫ mφ, the thermally-averaged decay rate for φ → f f in the lab frame is 〈Γφ〉 ≃(ǫ2/3)

∑

f (Q2fN

fc )αEMmφ(mφ/T ), where we have averaged over three φ polarizations, Qf and

Nfc are the standard model charge and number of colors for fermion f , the sum is over all

fermions with mass mf < mφ/2, and the last factor of mφ/T is from time dilation. Note thatthe decay process is inefficient at early times and high temperatures, but as T drops, 〈Γφ〉increases and H decreases, and so the φ particles may come into thermal equilibrium with

the the standard model. The expansion rate is H ≃ 1.66 g1/2∗ T 2/mPl; at freezeout, g∗ ≃ 100.

Requiring 〈Γφ〉 >∼ H at Tf ≃ mX/25 yields ǫ >∼ 3× 10−6 for mφ = GeV and mX = TeV.If the above constraint is not satisfied, then the hidden sector may be at a different

temperature from the standard model, modifying the dark matter freezeout. Such scenarioshave been considered in Ref. [37], and the difference in temperatures has a significant impact

7

on the dark matter relic density. These considerations may be important for some models,but here we assume that Eq. (18) is satisfied and mφ ≪ Tf ≃ mX/25, so that the φ particlesare in thermal contact with the standard model at freeze out and their number density isclose to the thermal prediction.

We note here that the calculation above does not guarantee chemical equilibrium. Forthe φ particles to produce X through φφ → XX , we need the tail of the φ distribution tobe populated thermally. To check this, we approximate the decay rate for φ with energiesof order mX as (ǫ2/3)

∑

f(Q2fN

fc )αEMmφ(mφ/mX). If this decay rate is larger than the

expansion rate, then the inverse process will also be in equilibrium and hence thermallypopulate the tail of the φ distribution. Comparing this to the expansion rate at Tf ≃ mX/25we obtain that

ǫ >∼ 1.5× 10−5

4∑

f Q2fN

fc

1

2[

GeV

mφ

]

[

mX

TeV

]3

2

. (18)

In setting the bound above, we have not considered scattering interactions with standardmodel fermions (and the resulting changes in kinetic energy for φ) because these processesare proportional to ǫ4 and hence considerably slower that the inverse decay process.

To summarize, for the φ particles to be in thermal equilibrium with the standard modelat freezeout requires that Eq. (18) is satisfied and mφ ≪ Tf ≃ mX/25. We will assume theseconditions hold in our analysis, but we note that they do not necessarily hold. In particular,if mφ ∼ Tf , the number of φ particles at freezeout is reduced, which reduces the X relicdensity, strengthening the bounds on Seff we determine below.

C. Kinetic Decoupling of Dark Matter

After freeze out, dark matter particles remain kinematically coupled to the thermal baththrough elastic scattering. Kinetic decoupling occurs later, when the momentum transferrate drops below the Hubble expansion rate. We define the kinetic decoupling temperatureTkd by Γk(Tkd) = H(Tkd), where Γk is the momentum transfer rate. It may be approximatedas

Γk ∼ nr〈σelvrel〉T

mX, (19)

where nr is the number density of the relativistic species in the thermal bath, and 〈σelvrel〉is the thermally-averaged cross section for elastic scattering between dark matter particlesand the relativistic particles in the thermal bath.

We first consider the model-independent elastic scattering off the hidden sector thermalbath of φ particles [38]. For T <∼ mφ, nφ is Boltzmann suppressed, and so this process willnot be able to maintain kinetic equilibrium. For T >∼ mφ, the thermally-averaged Xφ → Xφcross section is

〈σelvrel〉 ∼α2X

m2X

, (20)

the Thomson scattering cross section. Given the φ number density nφ ∼ T 3, the momentumtransfer rate is Γk ∼ α2

XT4/m3

X , and this is equal to H ∼ T 2/mPl at

T ∼[

m3X

α2XmPl

]1

2

= 0.43 MeV[

0.021

αX

] [

mX

TeV

]3

2

, (21)

8

where we have normalized the expression to typical parameters that give the correct relicdensity in the presence of Sommerfeld enhancement. Halo shape constraints require mφ

>∼30 MeV [2], and so for all physically viable and relevant parameters,

T φkd ∼ mφ . (22)

This is the temperature of kinetic decoupling from the hidden thermal bath. It is quitemodel-independent, as it assumes only the X-φ interactions required in all Sommerfeldenhancement scenarios, and so it provides a maximal value of Tkd in Sommerfeld scenarios.

Dark matter particles may also be kept in kinetic equilibrium through scattering off thevisible sector’s thermal bath. This is more model dependent, but as discussed in Sec. III B, ifφ is a U(1) gauge boson, it may mix with the standard model photon. After diagonalizing theφ-photon system, standard model particles with charge Q have hidden charge ǫQ, inducingnew energy transfer processes. The most efficient process is Xe → Xe scattering throught-channel φ exchange. For T <∼ me, Boltzmann suppression of the electron number densitymakes the interaction inefficient. At temperatures mφ

>∼ T >∼ me, the corresponding crosssection is [39]

〈σelvrel〉 ∼ǫ2αEMαXT

2

m4φ

. (23)

The momentum transfer rate Γk ∼ ǫ2αEMαXT6/(m4

φmX) is equal to the expansion rate attemperature

T ∼[

m4φmX

ǫ2αEMαXmPl

]1/4

, (24)

and so the resulting temperature of kinetic decoupling from the visible sector’s thermal bathis

T ekd(ǫ) ∼ max

me , 0.82 MeV

[

10−3

ǫ

]1

2[

mφ

30 MeV

] [

0.021

αX

]

1

4[

mX

TeV

]1

4

, (25)

where we have again normalized αX and mX to typical freeze out parameters, and addition-ally normalized mφ to its smallest possible value and ǫ to its largest allowed value. When

ǫ is near its maximal value, T ekd(ǫ) < T φ

kd, and so interactions with the visible thermal bathdelay kinetic decoupling. We denote the lowest possible kinetic decoupling temperature

T ekd ≡ T e

kd(ǫ = 10−3) , (26)

and will explore the dependence of our results on the temperature of kinetic decoupling byvarying it between its maximal value T φ

kd and its minimal value T ekd.

By crossing symmetry, the visible sector scattering interaction also implies an annihilationprocess XX → e+e−. In principle, this also enters the thermal relic density calculation.The cross section is 〈σanvrel〉 ∼ ǫ2αEMαX/m

2X , however, and the ǫ2 suppression makes this

subdominant for all relevant cases.

D. Velocity Distribution after Kinetic Decoupling

Before kinetic decoupling, dark matter particles have the same temperature as the thermalbath and a Maxwell-Boltzmann phase space distribution. Usually one assumes that the

9

phase space distribution remains Maxwell-Boltzmann after kinetic decoupling. However,this is not necessarily true in scenarios with Sommerfeld-enhanced annihilation, becauseslow particles annihilate with larger cross sections. This preferentially depletes the lowvelocity population and may distort the phase space distribution. In this case, the standardformulae used to compute the relic density of dark matter are not valid and one needs toexplicitly consider the effect of annihilations using the full phase space distribution.

As discussed in Refs. [2, 39–42], however, the φ field mediates self-scattering betweendark matter particles. If the self-scattering rate is higher than the Hubble expansion rate,the momentum exchanged in these self-interactions with be sufficient to maintain thermalequilibrium. The momentum transfer of particles interacting through Yukawa potentialshas been studied in Refs. [43, 44]. Although the authors of these studies were interestedin slow and highly charged particles moving in plasmas with screened Coulomb potentials,they approximated these potentials by Yukawa potentials, and so their results are exactlyapplicable in the current particle physics context. The numerical results of these studies forthe momentum transfer cross section are accurately reproduced by [43, 44]

σT ≈ 4π

m2φ

β2 ln(1 + β−1) , β < 0.1

σT ≈ 8π

m2φ

β2

1 + 1.5β1.65, 0.1 ≤ β ≤ 1000 (27)

σT ≈ π

m2φ

(

ln β + 1− 1

2ln−1 β

)2

, β > 1000 ,

where β = 2αXmφ/(mXv2rel).

Given these results, we may calculate the dark matter self-scattering rate and compareit to the Hubble expansion rate. The self-scattering rate is given by [2]

Γs =∫

f(~v1)f(~v2)nX (σT vrel)v2relv20

d3~v1d3~v2 , (28)

where nX is the dark matter density. This sets the time scale to change velocities by O(1).To find the rate to change the kinetic energy by O(1), one would divide the above rate by afactor of 3. However, such details are not important for the present calculation. H and Γs

are presented in Fig. 2 for two representative cases.We may define the self-scattering decoupling temperature Tnt by Γs(Tnt) = H(Tnt); after

Tnt, the dark matter velocity distribution may become non-thermal. To derive an approxi-mate expression for Tnt, note that to have the right relic abundance, the co-moving numberdensity of dark matter during the self-scattering epoch cannot be smaller than the co-movingnumber density at present. The dark matter number density nX may therefore be taken tobe

nX ∼ s(

nX

s

)

0= s

ΩDMρcmXs0

≃ s× 3.9× 10−13[

TeV

mX

]

. (29)

During the self-scattering decoupling epoch, typically β > 1000, and so σT ∼ κ/m2φ, where

κ ∼ π ln2 β ∼ 600 − 1000 for the parameters of interest. With this approximate expressionfor σT , we find

Tnt ∼ 20 keV[

mφ

250 MeV

] [

mX

1 TeV

]3

4

[

Tkd

250 MeV

]

1

4[

κ

800

]−1

2

. (30)

10

103 104 105 106 107 10810-32

10-30

10-28

10-26

10-24

10-22

10-20

10-18

10-16

10-14

Rat

e (G

eV)

mX/T

mX=100 GeV

X=0.00242m =0.25 GeV

Tkd=Tkd

104 105 106 107 10810-32

10-30

10-28

10-26

10-24

10-22

10-20

10-18

10-16

10-14

Rat

e (G

eV)

mX/T

mX=1 TeV

X=0.02m =0.25 GeV

Tkd=Tkd

FIG. 2: The self-scattering rate Γs (solid red) and the Hubble rate H (dashed blue) as functions

of x = mX/T for the values of mX , mφ, αX , and Tkd indicated.

This agrees well with our numerical results, and we use this approximate form with κ = 800in deriving our results below.

From both the numerical and analytical results, we see that the self-scattering rate effi-ciently preserves the thermal distribution to temperatures Tnt ∼ 10 keV, when dark mattervelocities are vrel ∼ T/(TkdmX)

1/2 ∼ 10−5. This is a low velocity, and in most cases, we willfind that it is sufficiently low that the impact of non-thermality after Tnt has a negligibleimpact on the thermal relic density. Note, however, that Tnt is not low enough to ensure athermal distribution down to vrel ∼ α4

X , where the Sommerfeld resonances cut off, and sovery close to resonances, the non-thermality will have an effect.

IV. RESULTS

A. Maximal Sommerfeld Enhancements

In this section, we present results for the maximal Sommerfeld enhancement. We havedefined several Sommerfeld factors, including S0, the original Sommerfeld factor for masslessφ given in Eq. (1), and S, the Sommerfeld factor generalized to massive φ, which includesresonances and is given in Eqs. (5) and (6). Here we also define the effective Sommerfeldenhancement factor

Seff =(σanvrel)0Snow

3× 10−26 cm3/s, (31)

where

Snow ≃ x3/2now

2√πN

∫ vmax

0Sv2rele

−xnowv2rel

/4dvrel , (32)

xnow ≡ 2/v20, and N = erf (z/√2 )− (2/π)1/2ze−z2/2 with z ≡ vmax/v0. Seff is the experimen-

tally relevant parameter, as it is the factor by which indirect fluxes are enhanced relative tothe case without Sommerfeld enhancement. Unlike the early Universe case, in the halo wehave to cut off the velocity integral at some maximum relative speed vmax, which is relatedto the escape speed from the local neighborhood. This maximum speed is a function of the

11

angle between the dark matter particles in the lab frame. However, we have checked thatsetting vmax to be equal to the escape speed vesc and integrating as in Eq. (32) makes adifference of less than 10%. The escape speed at about 10 kpc from the center of the halois estimated to be about 500 km/s to 550 km/s [45]. We use vesc = 525 km/s to derive ourresults, noting that 10% variations in vesc have a much smaller effect on Seff. We have as-sumed that the velocity dispersion tensor is isotropic with the 1-D velocity dispersion givenby v0/

√2. We use v0 = 210 km/s consistent with determinations of the circular velocity

(e.g., [45]). Inferring the dispersion from the circular velocity curve requires knowledge ofthe dark matter density profile. The value we use for v0 is consistent with a Navarro-Frenk-White density profile with a scale radius of about 20 kpc. Here again, we note that there areuncertainties in the value of v0 at the 10% level. This translates directely into an uncertaintyin Seff of about 10% if S ∝ 1/v and 20% around resonance where S ∝ 1/v2.

For comparison purposes, we also define

S0eff =

(σanvrel)0S0now

3× 10−26 cm3/s, (33)

where

S0now =

x3/2now

2√πN

∫ vesc

0S0v2rele

−xnowv2rel

/4dvrel (34)

is the effective Sommerfeld enhancement without resonances, but with the Sommerfeld effecton freeze out included.

To calculate the largest possible Seff for a given mX and mφ, we make a number ofoptimistic (Seff-maximizing) assumptions:

• We fix Tkd = T ekd. This delays kinetic decoupling as much as possible, keeps the

dark matter as hot as possible, and so reduces the Sommerfeld effect on freeze out,maximizing Seff.

• We fix xs in Eq. (13) by stopping the dark matter evolution at Tnt; that is, we neglectall annihilations that occur after the dark matter distribution becomes non-thermal.This is certainly optimistic, as the distribution will remain thermal for some timeand dark matter annihilations will continue, but this again minimizes the Sommerfeldeffect on freeze out, maximizing Seff.

• We require ΩXh2 = 0.114. This might appear to be too restrictive; after all, there is

no requirement that the observed signals arise from a particle that makes up all ofthe dark matter. However, if a flux arises from Sommerfeld-enhanced annihilation,it scales as n2

X〈σanvrel〉S ∼ α−1X , because its number density scales as nX ∼ ΩX ∼

〈σanvrel〉−1 ∼ α−2X and S ∼ αX . The flux can therefore always be increased by lowering

αX until ΩXh2 is all of the dark matter, and so choosing ΩXh

2 in fact maximizesindirect signals.

• We choose the maximal αX that yields ΩXh2 = 0.114. Roughly speaking, ΩXh

2

decreases as αX increases, and so typically, there is a unique choice of αX that yieldsthe correct ΩXh

2. In some cases with strong resonances, however, there are threechoices of αX that give the correct ΩX , as shown in Fig. 3. When we do a coarse scanthere is no guarantee that we always pick the solution with the largest value of Seff.Note, however, that the values of Seff only change by about 20–30% between the threeallowed solutions, and such variations do not modify our conclusions. The values ofαX that yield the correct relic density for given values of mX and mφ are given in

12

0.010 0.011 0.012 0.013 0.014

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Sef

f

Xh2

X

20

40

60

80

100

120

140

mX=580 GeVm = 1 GeVTkd=Te

kd

0.010 0.011 0.012 0.013 0.0140.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Tkd=Tkd

m = 1 GeVmX=580 GeV

Xh2

X

Seff

20

40

60

80

100

120

140

FIG. 3: The relic density ΩXh2 (solid red) and Seff (dotted blue) as a function of the fine-structure

constant αX for the fixed values of mX , mφ, and Tkd indicated and the observed value of ΩXh2

(dashed black). In most cases, there is a unique choice of αX that yields the correct ΩXh2 = 0.114

(left). In the presence of strong resonances, however, there are cases where three different choices

of αX all yield the correct ΩX (right). In these cases, Seff varies by about 20–30% between the

different solutions.

100 100010-3

10-2

10-1m =0.25 GeV

Tkd=Tekd

X

mX (GeV)100 1000

10-3

10-2

10-1

X

mX (GeV)

m =1 GeV

Tkd=Tekd

FIG. 4: The value of αX required to achieve a relic density of ΩXh2 = 0.114 as a function of the dark

matter particle mass mX (solid red) for mφ = 250 MeV (left) and mφ = 1 GeV (right). We also

plot the required αX (dotted blue) if Sommerfeld effects are neglected in the early Universe. The

tree level cross section (without Sommerfeld enhancement) is (σanvrel)0 = πα2X/m2

X . Because αX

varies over almost two orders of magnitude in this plot, the dips near resonance are not immediately

apparent.

Fig. 4. Note that close to a resonance, one needs extreme fine-tuning to avoid efficientannihilation in the early Universe and obtain the correct relic density.

In Fig. 5, we show the maximal values of Seff as a function of mX for mφ = 250 MeV and1 GeV. To understand the impact of resonances and the Sommerfeld effect on freeze outon these results, we also plot other Sommerfeld enhancements. S0, which includes neitherresonances nor freeze out effects, was used in Ref. [2]. We see that it is almost always

13

102 103100

101

102

103

Sef

f

mX (GeV)

m =0.25 GeV

Tkd=Tekd

102 103100

101

102

103

Seff

mX (GeV)

B

m =1 GeV

Tkd=Tekd

FIG. 5: The effective Sommerfeld enhancement factor Seff (solid red) as a function of mX for

mφ = 250 MeV (left) and 1 GeV (right) and the set of Seff-maximizing assumptions listed in

Sec. IVA. Also shown for comparison are S0 (dotted blue), the Sommerfeld factor without reso-

nances and neglecting the Sommerfeld effect on freeze out; S (dot-dashed blue), the Sommerfeld

factor with resonances but neglecting the Sommerfeld effect on freeze out; and S0eff (dashed red),

the Sommerfeld factor without resonances, but including the Sommerfeld effect on freeze out.

overestimates the maximal Sommerfeld enhancement. This is because, as evident in Fig. 5,the effect of Sommerfeld enhanced annihilation on freeze out is a significant suppression.This may be understood as follows. If Sommerfeld enhancement reduces the thermal relicdensity by a factor ζ , the tree-level cross section (σanvrel)0 must be reduced by ζ to keep ΩX

fixed. However, Seff ∝ α3X ∝ (σanvrel)

3/20 , and so reducing ΩX by ζ implies a reduction in Seff

by a factor ζ3/2. For example, for mX = 1 TeV, the maximal coupling is αX ≃ 0.021, andζ ∼ 1.5, consistent with the results of Refs. [29, 30]. Including the effect of freeze out heretherefore reduces S0 to S0

eff by a factor of (1.5)3/2 ∼ 2, a significant reduction.Adding resonances then produces oscillations about the S0 and S0

eff contours, with peakpositions given by Eq. (7). We see that, although resonances can significantly enhance theeffective Sommerfeld factor, these enhancements are significant only for low values of n; formφ

<∼ GeV, these lie at low mX , and the effect becomes negligible for mX>∼ 1 TeV.

In Fig. 5, we have fixed mφ to typical values considered in the literature. In Fig. 6, we plotupper bounds on Seff as a function of mφ for fixed values of mX . We see that the resonanceshave little impact for mφ

<∼ 1 GeV, but can produce enhancements by factors of 2 to 3 forlarger mφ. Large values of mφ

>∼ 1 GeV have been considered disfavored, however, as theyeliminate the kinematic suppression of anti-proton fluxes, which are typically considered tobe consistent with astrophysical backgrounds.

What is the effect on Seff of deviating from the Seff-maximizing assumptions? The effectof the choice of Tkd is highly sensitive to how close one is to a resonance. Far from aresonance, there is little sensitivity to Tkd values between T φ

kd and T ekd. As one approaches a

resonance, however, Seff may vary by ∼ 10% or more. In particular, we found regions closeto resonances for mφ ∼ GeV where Seff changed by about 30%. This behavior is shownin Fig. 7. If one is not very close to a resonance, the choice of xs has a small impact. Incontrast, as discussed above, the assumption of ΩXh

2 = 0.114 has a large effect; for otherchoices, the maximal value of Seff scales as (ΩXh

2)3/2. Finally, as can be seen in Fig. 3,

14

1 2 3 4 5

100

101

102

mX=100GeV

mX=3TeV

Sef

f

m (GeV)

mX=1TeV

mX=300GeV

Tkd=Tekd

FIG. 6: The effective Sommerfeld enhancement factor Seff as a function of mφ for the mX indicated

and the set of Seff-maximizing assumptions listed in Sec. IVA.

200 400 600 800 1000

50

100

Seff

mX (GeV)

m =0.25 GeV

200 400 600 800 1000

50

100

150

200

250

Sef

f

mX

m =1 GeV

FIG. 7: The effective Sommerfeld enhancement factor Seff for Tkd = T ekd (solid red) and Tkd = T φ

kd

(dashed blue), for mφ = 250 MeV (left) and 1 GeV (right). All other Seff-maximizing assumptions

have been made. Near resonances, Seff is sensitive to the temperature of kinetic decoupling.

different choices of αX also decrease Seff ∝ α3X by ∼ 20− 30%.

For typical parameters that are not very close to a resonance, then, we expect thatassumptions different from those listed above would produce a small decrease in Seff whenaway from resonances. Note, however, that very near a resonance, all of these choices becomevery important, as we now discuss.

15

102 103 104 105 106 107 108 109 101010-35

10-33

10-31

10-29

10-27

10-25

10-23

10-21

10-19

10-17

10-15

Tkd=Tkd

m =0.25 GeVX=0.0112

mX=500 GeV

Xh2=0.11 (Ts=1 eV)Xh2=0.11 (Ts=Tnt)

mX/T

Rat

e (G

eV)

102 103 104 105 106 107 108 109 101010-35

10-33

10-31

10-29

10-27

10-25

10-23

10-21

10-19

10-17

10-15

Xh2=0.000161 (Ts=1 eV)Xh2=0.022 (Ts=Tnt)

Tkd=Tkd

m =0.848 GeVX=0.0112

mX=500 GeV

Rat

e (G

eV)

mX/T

FIG. 8: The annihilation rate Γan = nX〈σanvrel〉 (solid red) and the Hubble rate H (dashed

blue) as functions of x = mX/T for the values of mX , mφ, αX and Tkd indicated. Without

resonant Sommerfeld enhancement (left), Γan < H after freeze out; but with resonant Sommerfeld

enhancement (right), Γan may become comparable toH at late times, leading to chemical recoupling

and an new era of annihilation. Also indicated on the plots are the resulting values of ΩXh2 that

result from assuming annihilations up to a temperature of Tnt when the dark matter distribution

function is no longer able to maintain kinetic equilibrium, as well as a lower temperature of 1 eV.

B. Resonances and Chemical Recoupling

Very near a resonance, the annihilation cross section becomes sensitive to details of thedark matter’s evolution to low velocities at late times. The effective Sommerfeld enhance-ment then becomes highly sensitive to details of Tkd, the velocity distribution of dark matterafter Tkd, and the cutoff of resonant enhancements. As this is far from generic, we do notpresent details of these dependences, but we note that, contrary to naive expectations, Seff

is not maximized by sitting exactly at resonance. In fact, exact resonances lead to extremelyefficient annihilation in the early Universe and minimize current indirect signals.

Resonant Sommerfeld enhancement’s effect on freeze out can also be so large that it leadsto the intriguing phenomenon of chemical recoupling. This is illustrated for two represen-tative cases in Fig. 8. Without resonant Sommerfeld enhancement, dark matter freezes outand remains frozen out, but with resonant enhancement, dark matter may melt back in atlate times, or chemically recouple, leading to a second era of efficient annihilation.

The phenomenon of chemical recoupling in the context of Sommerfeld enhancements maybe understood as follows. The Hubble parameter scales as H ∼ T 2. The annihilation rateis Γan = nX〈σanvrel〉. The dark matter density scales as nX ∼ T 3, and 〈σanvrel〉 scales as

v−1rel ∼ T

−1/2X off resonance and v−2

rel ∼ T−1X on resonance. Before Tkd, TX = T , and so the

annihilation rate cannot grow relative to the expansion rate. However, after Tkd, TX ∝ T 2,and so Γan ∼ T . At late times, then, the annihilation rate decrease slowly enough to becomecomparable to or greater than the expansion rate. The dark matter then melts back in, andthere is a new era of annihilation. In these cases, the relic density is very sensitive to thetemperature at which we stop including the annihilation process. As our default case, wehave been conservative and stopped the annihilation process when the self-interactions ofdark matter are no longer able to maintain kinetic equilibrium. Annihilations will proceed

16

500 1000 1500 2000 2500 3000 3500 4000101

102

103

Sef

f

mX (GeV)

PAMELA

Fermi

m =0.25 GeV

Tkd=Tekd

Ts =Tnt

FIG. 9: The maximal effective Sommerfeld enhancement factor Seff compared to data for force

carrier mass mφ = 250 MeV. The cross-hatched region is excluded by requiring consistency with

the thermal relic density and adopting all of the Seff-maximizing assumptions listed in Sec. IVA.

The red and green shaded regions are 2σ PAMELA- and Fermi-favored regions for the 4µ channel,

and the best fit point is (mX , Seff) = (2.35 TeV, 1500) [20].

beyond this temperature; however, we cannot assume a Maxwellian distribution for the darkmatter particle, because the annihilations preferentially deplete the low momentum tail. Asan example, we show in Fig. 8 what happens when we allow the annihilation process toproceed down to 1 eV. The relic density is essentially negligible for the case where we havechemical recoupling, whereas it is unchanged for the case away from resonance.

V. COMPARISON TO PAMELA AND FERMI

A. Maximal Enhancements and Best Fit Parameters

In Figs. 9 and 10, we compare the maximal Seff presented in Fig. 5 to the boost factorsrequired to explain PAMELA and Fermi data. The PAMELA and Fermi regions, derivedin Ref. [20], assume an isothermal halo and a 250 MeV force carrier that decays with 100%branching ratio to µ+µ−, leading to annihilations XX → µ+µ−µ+µ−. The fits are insensitiveto mφ, provided mφ ≪ mX . The best fit is for (mX , Seff) = (2.35 TeV, 1500), and the regionsare 2σ contours relative to the best fit parameters. Results for other halo profiles, otherparticle physics models, and other final states have also been considered [20, 21, 46]. Thesestudies find that the 4µ final state provides a better fit than all other considered final states,but see Sec. VD for a discussion of this issue.

Despite choosing Tkd and all other parameters to maximize Seff, we find that the possiblevalues of Seff fall short of explaining the PAMELA and Fermi excesses. For example, atmX = 2.35 TeV the maximal Sommerfeld enhancement is Seff ≈ 100, a factor of 15 belowthe best fit value. For lower mX , the data may be fit with smaller Seff, but the relic density

17

500 1000 1500 2000 2500 3000 3500 4000101

102

103

m =1 GeV

Tkd=Tekd

Ts =Tnt

Seff

mX (GeV)

PAMELA

Fermi

FIG. 10: The maximal effective Sommerfeld enhancement factor Seff compared to data for mφ =

1 GeV. The cross-hatched and shaded regions are as in Fig. 9. The PAMELA- and Fermi-favored

regions are for mφ = 250 MeV, but do not change appreciably for mφ = 1 GeV [21] in the case

that only the 4µ channel is included.

bounds are also stronger; for example, for mφ = 250 MeV and mX = 0.1, 0.3, and 1 TeV,the maximal values of Seff are 7, 30, and 90, respectively.

B. Astrophysical Uncertainties

Our analysis so far has assumed standard astrophysics and astroparticle physics. Toenable Sommerfeld enhancements to explain PAMELA and Fermi, one may consider uncer-tainties in these assumptions.

The annihilation signal is sensitive to the square of the dark matter density ρX in thelocal neighborhood. The best fit regions given in Ref. [20] assume the conventional valueρX = 0.3 GeV/cm3, which has traditionally been considered to be uncertain up to a factor of2 [47]. More recently, studies have tried to refine the calculation of the uncertainty in the localdensity of dark matter. One study found ρX = 0.389±0.025 GeV/cm3 [48], using an ensembleof kinematic tracers to constrain a standard galactic model with a spherical halo, stellar andgas disk and stellar bulge. Another recent study [49] finds ρX = 0.2 − 0.4 GeV/cm3. Athird study Taylor-expanded the local rotation velocity curve to first order and used localkinematic tracers and an estimate of the baryonic contribution to the rotation curve tofind ρX = 0.430 ± 0.113 ± 0.096 GeV/cm3 [50], where the first uncertainty is from theslope of the circular velocity V (r) at the Sun’s radius and the second is from uncertaintyin distance of the Sun from the galactic center. The three studies are consistent with eachother, but the first one finds significantly smaller uncertainty in the value of ρX . If we usethe results of Ref. [48], then the best fit regions plotted in Figs. 9 and 10 shift down by afactor of (0.4/0.3)2. Of course, it is not appropriate to simply scale up the signal by some ρ2Xrequired to fit the positron and electron data; in a proper analysis, one should include the

18

appropriate likelihood for ρX and ΩXh2. We have also discussed some of the uncertainties

in the velocity distribution function and their effects in Section IVA, and, as with ρX andΩXh

2, these uncertainties should be included in the full likelihood analysis.Another effect that was pointed out recently [51] is that the positrons produced within

the unseen bound subhalos out in the Milky Way dark matter halo could propagate tothe local neighborhood and contribute to the PAMELA and Fermi signal. For the bestfit point, this study found that about 30% of the positrons in PAMELA could be due toannihilations in the subhalos [51]. Taking ρX = 0.4 GeV/cm3 and an enhancement factorof 30% from subhalos, the discrepancy is reduced from a factor of 15 to a factor of 4. Wesee that these changes are insufficient to reach the best fit regime, even if all the parametersare simultaneously pushed in the most optimistic direction. A more troubling aspect ofthis calculation is however the fact that the subhalos considered were those resolved by theVia Lactea II simulation. This implies that the much larger contribution from the lowermass subhalos would over produce the signal. Appealing to subhalos to O(1) changes to therequired Seff is therefore unmotivated.

Finally, we may also appeal to small scale structure in the immediate local neighborhoodto enhance the positron signals. Such a contribution is largely unconstrained by data, butpresent simulations, even without the deleterious effect of the disk of stars, do not predictlarge enhancements from such an effect [52]. A recent study [53] using the Via Lactea IIsimulation and including the Sommerfeld effect for annihilation in the local clumps foundthat the probability of finding a subhalo close enough to make an O(1) effect on the e+e−

flux above about 100 GeV is about 4% and that this contribution could be about 15% if theVia Lactea II subhalo mass function is extrapolated down to lower (unresolved) masses. Twopoints are worthy of further note here. First, these subhalo results are going to be cruciallyaffected by the disk. The zeroth order expectation is that the gravity of the stellar disk willnot allow such subhalos to survive or even if they do, truncate it significantly. Second, forsmall mφ and near resonances self-interactions are important [2, 41] and that could modifythe internal structure of the subhalo as well as the mass function.

A more complicated set of astrophysical uncertainties is related to astrophysical back-grounds and the signals themselves. For example, the required Sommerfeld enhancementmay be reduced if conventional astrophysical backgrounds have been under-estimated, cos-mic ray propagation models are modified, pulsars contribute significantly to the positronflux, or the PAMELA e+ sample includes some proton contamination. Any of these effectscould reduce the required dark matter signal, but they could also plausibly eliminate theneed for dark matter altogether, as has been argued in numerous studies [54–59]. It is there-fore typically unclear what the motivation would be for assuming that such effects accountfor part of the signal but not all of it, independent of a desire to leave room for a darkmatter signal.

A possible exception to the above argument is the possibility of adjusting cosmic raypropagation models to match the Fermi spectrum. Such adjustments could remove the needto explain Fermi, without changing the PAMELA excess significantly [60]. As shown inFig. 10, for mX

<∼ TeV and mφ>∼ 1 GeV, the allowed Sommerfeld enhancements may

be in marginal agreement with PAMELA data. This scenario is, however, problematic. Acontribution to the e± flux at about 100 GeV to explain the observed PAMELA positronfraction implies a modification to the e± flux observed by Fermi. For example, a 500 GeVdark matter explanation for PAMELA with the correct relic density is marginally possiblefor mφ = 1 GeV according to the 4µ fits of Ref. [21] as shown in Fig. 10. This would,

19

however, imply a drop in the e± flux beyond 500 GeV, which is not seen by Fermi, a pointstressed in previous works [21, 46]. Analysis of such a scenario with modified propagationmodels would be another avenue for further study.

C. Cosmic Microwave Background Bounds

In this section, we include bounds from the effect of residual annihilation at last scatteringon the ionized electron fraction and therefore on the cosmic microwave background (CMB)power spectrum. The CMB bound may be written as [61, 62]

S|v=0 =12

ǫφ[

1− cos(√

24/ǫφ)] <

120

f

(

mX

TeV

)

, (35)

where f is the average fraction of the energy produced in annihilation that reionizes Hydro-gen between redshifts of 800 and 1000. For the e± final state, f ≃ 0.7, while for the µ± finalstate, f ≃ 0.25 [62]. The left hand side S|v=0 is the saturated Sommerfeld enhancementobtained by setting v = 0 in Eq. (6). This is sufficient to approximate the Sommerfeld en-hancement during recombination, unless one is right on top of a resonance, which is highlydisfavored by this bound.

Equation (35) is not a monotonic function of αX . To impose this bound, we consider aregion in αX that is bounded by values 10% larger and smaller than the value of αX dictatedby the relic density bound of ΩXh

2 = 0.114. We find that large resonances are no longerallowed by the CMB bound as a comparison of Fig. 10 and Fig. 11 will show.

D. Force Carrier Decay Channels

The best fit regions of the (mX , Seff) plane shown in Figs. 9 and 10 assume that the forcecarrier decays solely to muons, leading to the XX → φφ → 4µ annihilation channel. The 4µfinal state has been shown [20, 21, 46, 63] to be the final state that yields the best fit of allconsidered so far, as it leads to smooth e± distributions that can simultaneously explain bothPAMELA and Fermi. At the same time, this is an inefficient mode, as significant energyis lost to neutrinos. To ameliorate the discrepancy between the required Seff to explain theanomalies and the maximal Seff allowed by the relic density constraint, one might considerother channels, which could potentially allow similar contributions to the e± spectrum withlower Seff and at lower mX .

The existence of other decay channels is theoretically well-motivated and there are manypossibilities. If the decay φ → µ+µ− is possible, the decay φ → e+e− is also kinematicallyallowed. For mφ ∼ 1 GeV, other decay modes, such as φ → ππ,KK, are also possible. Asan example, if φ decays are controlled by its kinetic mixing with the standard model photon,then there will also be e± and π± channels with roughly similar branching ratios, dependingon mφ [64]. Alternatively, φ particles may decay preferentially to the heaviest availablestates if they are coupled through standard model Yukawa couplings, or may have otherinteresting dynamics. In fact, in full generality, the φ particles may decay not only to pairsof standard model particles, but also through final states involving hidden Higgs bosons andother hidden particles, leading, for example, to XX annihilations to 6 or 8 particle finalstates.

20

500 1000 1500 2000 2500 3000 3500 4000mX (GeV)

101

102

103

S eff

FIG. 11: The maximal effective Sommerfeld enhancement Seff (black solid curve) after including

bounds from CMB (see Sec. VC) compared to fits for mφ = 1 GeV for exclusive 4e and 4µ modes.

The dark red (vertically hatched) region is the 99% contour for the 4µ mode and the light red

(horizontally hatched) contour is for the 4e mode [21]. The best fit regions for the 4e mode are at

lower Seff and mX , but the best fit 4e mode point has a significantly lower likelihood than the best

fit 4µ mode point [21] for a fixed cosmic ray propagation model.

Studies so far have found that the 4µ mode is a better fit than other channels [21, 63].A precise quantitative statement requires inclusion of correlations through covariance ma-trices for each experiment, and these are not available. Given this fundamental uncertainty,conclusions about the µ mode and other channels are far from firm and require additionalassumptions.

In Ref. [63], the 4e mode is shown to have a χ2/dof that is larger by about ∼ 1 comparedthe 4µ mode. These values of χ2 are computed by assuming that the data points areuncorrelated and allowing for the spatial diffusion parameter to vary. Other results obtainedby fixing the diffusion parameter seem to agree qualitatively with this result [21]. In fact,marginalizing over mX and other parameters, Ref. [21] found that the possibility of mixedmodes in the ratio of 4e : 4µ = 1 : 1 is excluded at 99.9% CL.

Despite these results, we may consider what impact other modes might have, especiallyin light of the fact that there is a lot of freedom in the cosmic ray propagation modelparameters. As an illustration, we consider the best fit regions from Ref. [21] for both the4e and 4µ modes separately. We plot these best fit regions with the relic density and CMBupper limits on Seff in Fig. 11. The best-fit point for the 4e mode is at lower Seff and mX

and the edge of the region is in marginal agreement with the relic density constraint. Ofcourse, as discussed above, the best-fit point for the 4e mode has a much lower likelihoodthan the best-fit point for the 4µ mode, according to Ref. [21]. For a 1:1 ratio, the best fitpoint would move to X masses somewhere between 1.5 and 2 TeV. This is disfavored by therelic density requirement even if we set the local dark matter density to be 0.4 GeV/cm3.

For the reasons noted above, all of these conclusions are subject to many uncertainties.The present conclusion in the literature seems to that significant deviations from the pure

21

4µ channel do not provide a good fit to data. One way forward would be to redo fits ofthe kind discussed in literature, allowing for variations in cosmic ray propagation modelparameters to see if this conclusion holds up in general. In any case, a careful statisticalanalysis is required before appeals to alternative channels alone may be considered as a wayto allow Sommerfeld scenarios to explain the data.

E. Bound States

For the case of attractive forces, pairs of dark matter particles may form bound statesby radiating force carrier particles if the force carrier mass mφ is below the binding energyα2XmX/4 [18, 65, 66]. As in the case of positronium, when this is kinematically possible,

annihilation through bound states is preferred. In the limits mφ ≪ α2XmX/4 and v ≫

mφ/mX , the annihilation rate enhancement is about 20αXvφ(3 − v2φ)/(2v) for fermions,where vφ is the velocity of the final state φ particle when the bound state forms [18]. Thisis roughly a factor of 7 larger than S0 when vφ ≃ 1.

Unfortunately, such bound state effects do not help Sommerfeld enhancement modelsexplain the PAMELA and Fermi data for two reasons. First, bound state formation iskinematically forbidden in large parts of parameter space that yield the best fits to data,and the large factor of 7 enhancement alluded to above is not realized for most of theparameter space plotted in Fig. 10. To see this, we first need to compute the value of αX

required to get the right relic density. Now, bound states may form in the early Universewhen they are energetically viable, that is when v < αX , and hence affect the relic densitycalculation. For simplicity we neglect this effect here, and use the same αX values fromFig. 4, which leads to a conservative bound. Then, for mφ = 0.25 GeV (1 GeV), the boundstate enhancement formula quoted above is valid only for mX ≫ 1.5 TeV (2.5 TeV). Theseinequalities are not satisfied for parameters that are most promising for fitting the data(mφ

>∼ 0.25 GeV and mX<∼ 4 TeV).

Second, if bound states form, they are likely to be in conflict with bounds from theCMB. To see this, let’s saturate the bound state enhancement at v ∼ mφ/mX and esti-mate the enhancement at recombination as 20αXmX/mφ. This should be compared to theCMB bound of (120/f)(mX/TeV) ≃ 500 (mX/TeV), where we have set f = 0.25 to beconservative. To satisfy the CMB bound, then, we require αX < (mφ/GeV)/40. This isgenerically violated if αX is fixed by the relic density. For example, for mφ = 1 GeV, thebound implies αX < 0.025. Requiring the correct relic density then implies mX < 1 TeV.However, we need mX > 2.5 TeV for bound states to form with mφ = 1 GeV. The situationis no different with mφ = 0.25 GeV. Thus our saturation approximation above (for boundstate enhancement) suggests that regions of parameter spaces where bound states can formare generically in conflict with the CMB.

F. Non-minimal Particle Physics Models

A potentially more promising direction is to construct more complicated particle physicsmodels to achieve Sommerfeld enhancements that are large enough to explain PAMELA andFermi. Such models will generically include additional annihilation channels. We begin bydiscussing the impact of these channels in general, and then examine various strategies onemight explore to achieve larger Seff.

22

1. Additional Annihilation Channels

As discussed above, to maximize Seff, we have considered only XX → φφ annihilation,leading to the typical tree-level cross section estimate of Eq. (2). Even in the simplestmodels, however, one may have additional annihilation channels. For example, if φ is a U(1)gauge boson with mass generated by spontaneous symmetry breaking through the Higgspotential 1

2λ(|H|2 − v2)2, where H is a complex scalar, there is a physical hidden Higgs

boson h with mass mh ∼√2λ v, where v is related to the φ mass by mφ =

√8παX v. For

perturbative λ, mh<∼ 10mφ; h may be much lighter than φ, but it cannot be much heavier.

The hidden Higgs boson therefore cannot be decoupled either dynamically or kinematically,and there is necessarily an additional annihilation channel XX → φh. In the non-relativisticlimit, and assuming there is no Yukawa coupling hXX , the cross section is

σ(XX → φh)vrel =πα2

X

m2X

|pφ|mX

4m4X + 8m2

Xm2φ

(4m2X −m2

φ)2

, (36)

where

|pφ| =[(4m2

X − (mφ +mh)2) (4m2

X − (mφ −mh)2)]

1/2

4mX. (37)

In the limit mφ, mh ≪ mX ,

σ(XX → φh)vrel =1

4

πα2X

m2X

. (38)

Additional annihilation channels and their impact on relic densities in Sommerfeld enhancedmodels have also been considered in, for example, Ref. [27].

To roughly accommodate such modifications, we may parameterize the effect of additionalannihilation channels by assuming a tree-level annihilation cross section

(σanvrel)0 = kπα2

X

m2X

, (39)

where k is a constant. If the new structures do not significantly modify the Sommerfeldenhancement S, the desired relic density will be achieved by modifying αX → αX/

√k,

moving the positions of resonances and reducing the maximal Seff by the factor√k. In

general, Sommerfeld enhancement in the early Universe is important and the reduction inmaximal Seff is smaller. The effects of varying k are shown in Fig. 12.

Of course, in general, new annihilation channels may have different couplings αX , differ-ent kinetic decoupling features, and different Tnt, all leading to different Sommerfeld effects,resonances in different places, and many other effects that cannot be captured by a singleextra parameter k. The availability of additional annihilation channels also has a compli-cated effect on the signal. For example, for the case of decays to Higgs bosons and assumingh → 4l through real or virtual φ pairs, the hφ mode leads to six or more leptons. Thischannel changes the energy spectrum of positrons that would be seen by PAMELA andFermi and thus requires new fits to the data. The full analysis is therefore complicated,and requires a dedicated study incorporating all of these effects into the freeze out analysis.The main point is that non-minimal models will typically include additional hidden sectorstates and additional annihilation channels, generically reducing the maximum possible Seff

consistent with thermal relic density constraints.

23

102 103100

101

102

103

m =0.25 GeV

Tkd=Tekd

Seff

mX (GeV)

B

k=1

k=4

102 103100

101

102

103

m =1 GeV

Tkd=Tekd

Seff

mX (GeV)

B

k=1

k=4

FIG. 12: The maximal effective Sommerfeld enhancement factor Seff for k = 1, 4, where the

tree-level annihilation cross section is (σanvrel)0 = kπα2X/m2

X , for mφ = 0.25 GeV (left) and

mφ = 1 GeV (right). The effects scale roughly as 1/√k, with the deviation from 1/

√k scaling due

to the Sommerfeld effect in the early Universe.

2. Multi-state Dark Matter

Two-state dark matter models have received considerably attention recently. For example,“inelastic dark matter” has been motivated by the DAMA NaI signal [67, 68] and “excitingdark matter” has been proposed to explain the INTEGRAL/SPI excess towards the galacticcenter [69]. They have also been proposed as explanations of the PAMELA excess and theWMAP haze towards the galactic center [38, 70]. In these models, the stable dark matterstate is accompanied by a more massive unstable state, which is separated from the stablestate by a mass gap that is chosen to fit the data and is much smaller than the dark mattermass. The presence of this almost degenerate excited state significantly complicates theearly Universe freeze out analysis. The Sommerfeld enhancements are also different [25]. Tocompute the relic density, one would need to include the additional annihilation channelsand the two-state dark matter Sommerfeld enhancement in a self-consistent early Universefreeze out calculation.

Dark matter models with multiple stable dark matter states have also been proposed toexplain the PAMELA and Fermi excesses and the DAMA/LIBRA signal [71–73]. Thesemodels provide an interesting avenue to reduce the required Sommerfeld enhancement toexplain both the PAMELA and Fermi excesses [73]. This is because the heavier (∼ TeV)component contributes mainly to the Fermi excess, whereas the lighter (∼ 100 GeV) darkmatter particle contributes predominantly to the PAMELA positron excess. The requiredSommerfeld enhancements are about 100 for the TeV state.

Such enhancements may appear to be consistent with our relic density constraint to withina factor of 2, but the relic density calculation that we have presented here is not completein these scenarios. In the simplest such models [73], the two states couple to the dark gaugeboson with the same charge. As stated, however, these models are ruled out by the directdetection limits, because the Xq → Xq cross section due to the kinetic mixing would betoo large for ǫ > 10−6 [17, 36]. The way out of this constraint is to specify that each stateis further split into a metastable and stable state with very small mass gap and that the φ

24

particles only have off-diagonal couplings, so that kinetic energy larger than the mass gap isrequired to initiate Xq → X∗q, where X∗ denotes the excited state of X [17]. These massgaps may then be chosen to be ∼ O(MeV) or∼ O(0.1 MeV) to explain INTEGRAL/SPI andDAMA/LIBRA signals respectively [73]. The relic density calculation is now significantlymore complicated. Additional annihilation channels, for example, annihilations between theheavy and light states and annihilation to hφ, and the presence of an almost degenerate statemust be taken into account in the early universe calculation. The Sommerfeld effect for thelighter state is generically important because the coupling to the dark gauge boson would beset by the relic density constraint for the heavy particle, and so αX ∼ 0.02, much larger thanthe required αX for thermal dark matter with mass ∼ 100 GeV. The new channels, suchas the hφ channel, also modify comparisons to PAMELA and Fermi as these new channelsmay contribute to the e± flux with a different energy spectrum. These issues are beyondthe scope of the present work.

Multi-state dark matter theories may also have significantly suppressed k. For example,in the non-Abelian models of Ref. [74], values of k as small as 0.14 are possible, and thesehave recently been claimed to provide marginally consistent explanations of the cosmic rayanomalies [75]. In these studies, however, the irreducible annihilations to Higgs bosons havenot been included, and the Sommerfeld effect on freeze out has been neglected. Given thelow value of k, these changes to the annihilation cross section may be important. It wouldbe interesting to see if the consistency remains after a complete analysis.

To summarize, finding a model with more degrees of freedom that requires smaller booststo fit the PAMELA and Fermi data does not by itself imply an improvement, as the boundson Seff will generically also be stronger. Generalizations of the freeze out analysis describedhere are required to determine if these scenarios may self-consistently explain PAMELA andFermi data.

3. Very Heavy Force Carriers with mφ ≫ 1 GeV

One reason the maximal Seff is somewhat limited is that the resonances are not significantfor mφ

<∼ 1 GeV and mX>∼ 1 TeV. For larger mφ, however, the resonances move to larger

mX , as evident from Eq. (7). Generically, for mφ > 2 GeV, φ decay may produce anti-protons. However, if such contributions to the anti-proton spectrum may be otherwisesuppressed or accommodated [76], such large mφ may allow one to improve the consistencyof Sommerfeld-based explanations of the PAMELA and Fermi data.

As an example, in Ref. [19], a leptophilic model was constructed where the anti-proton fluxis suppressed by dynamics, freeing the force carrier mass to be much larger than ∼ GeV.For mφ ∼ 10 GeV and mX ≈ 800 GeV, the authors noted that resonant Sommerfeldenhancement could enhance signals with enhancement factors as large as S ∼ 1000. However,the study of Ref. [19] did not include the Higgs annihilation channel, and did not include theimpact of Sommerfeld enhancement on freeze out. Note also that, as discussed in Sec. III B,for very largemφ near the freeze out temperature, φ particles may not be in complete thermalequilibrium with the standard model, modifying the freeze out analysis. Nevertheless, suchpossibilities certainly merit further study. Note, though, that these models are inaccessibleto searches for GeV forces.

25

4. Running αX

In all of our analysis above, we have assumed that the running of the gauge couplingin the dark sector is not significant. Although this is certainly a good approximation forthe minimal Abelian model we have considered, the situation is much more complicatedfor non-Abelian models. It has been recently pointed out that a non-Abelian hidden sectorcould explain both PAMELA and Fermi, while resulting in the correct relic density [77].This work notes that the Sommerfeld effect is a soft process and the enhancement factorS is determined by the coupling at scale mφ, whereas for the tree-level cross section thecoupling must be evaluated at mX . For a non-Abelian theory where the coupling constant issmaller at higher energies, this could significantly relax the tension between the relic densityconstraint and the requirements for explaining PAMELA and Fermi.

However, this analysis [77] did not include the Sommerfeld effect on the annihilation in theearly Universe. This would be particularly important because the Sommerfeld enhancementis now much larger even in the early Universe. The model also includes a Higgs doubletH and the dark matter annihilation XX → HZ must also be included. The stability ofthe Higgs sector is a major concern for these models because the Higgs annihilation crosssection is too small to suppress their relic density [77]. The decay lifetimes that have beencomputed [77] are long (∼ 106 s or longer) and could destroy the successful predictions ofBig Bang nucleosynthesis (BBN) because of the injection of electromagnetic energy [78, 79].Given these issues, it is not clear that these models can successfully satisfy the relic densityconstraints, but future work addressing these points could be an interesting way forward toexplain PAMELA and Fermi excesses using Sommerfeld-enhanced annihilation.

We note that the arguments regarding the decay of the hidden sector Higgs boson arequite general. The hidden Higgs boson should decay to standard model particles, otherwiseit would contribute to the dark matter abundance. In doing so, the decays should notdestroy the successful predictions of BBN.

VI. CONCLUSIONS

In this work, we have investigated scenarios with Sommerfeld-enhanced annihilation ofthermal dark matter and determined the largest possible indirect signals, subject to the mostbasic of constraints, namely, that the thermal dark matter have the correct thermal relicdensity. In addition to their potential impact on cosmic e± spectra, such scenarios may alsopredict observable effects in the cosmic microwave background and the diffuse extragalacticγ-ray background, excess γ-ray, neutrino and radio wave emission from the galactic center,and excess γ-rays from dwarf spheroidal satellites of the Milky Way and extended regionscentered on the galactic center [21, 46, 61, 62, 80–88]. The absence of each of these potentialsignatures provides significant constraints. The bounds derived here are complementary tothose, in that these bounds are derived by studying not the annihilation products, but theproduction. These results are therefore model-independent and robust, in that they assumeonly that the dark matter is produced by thermal freeze out, which is the central assumptionmotivating the consideration of Sommerfeld enhancements in the first place.

This analysis included the possibility of resonant Sommerfeld enhancement and the effectof Sommerfeld enhancement in the thermal relic density calculation. The possibility ofSommerfeld enhancement at low velocities, with annihilation cross sections proportional tov−1rel , or even v−2

rel on resonance, implies that the thermal relic density is sensitive to the dark

26

matter density’s evolution long after conventional freeze out at Tf ∼ mX/25. As a result,the relic density depends on many aspects absent in conventional scenarios. These includethe cutoff of resonant Sommerfeld annihilation, the equilibration of force carrier particles,the temperature of kinetic decoupling Tkd, and the efficiency of self-scattering to thermalizethe dark matter velocity distribution after Tkd.

These dependences may be very important. A non-generic but intriguing example is that,on resonance, dark matter freezes out but may then later melt in, chemically recoupling fora second era of efficient annihilation. As a result, exact resonances suppress rather thanenhance indirect signals, in contrast to naive expectations. More generally, the Sommer-feld enhancement of freeze out implies that smaller tree-level annihilation cross sections arerequired to give the correct relic density, and so reduces the effective Sommerfeld enhance-ment of indirect signals. We have found reduction factors of a few, with several ∼ 10− 30%variations possible, depending on the value of Tkd and other freeze out parameters.

For the minimal scenario analyzed here, and adopting the most optimistic assumptionsto maximize indirect signals, the largest possible values of Seff are 7, 30, and 90 for mφ =250 MeV and mX = 0.1, 0.3 and 1 TeV, respectively. As shown in Figs. 9 and 10, suchenhancements fall short of explaining the PAMELA and Fermi cosmic e± excesses. For thebest fit point, as discussed in Sec. V, this discrepancy is over an order of magnitude andcannot be eliminated by appeals to enhanced local dark matter density or boosts from smallscale structure. Bound state effects are also unable to eliminate the inconsistency, as theyare typically insignificant in the parameter regions that may potentially explain the cosmicray anomalies, and are in any case stringently constrained by the CMB. Along with theastrophysical uncertainties discussed in Sec. V and the complementary constraints from thecosmic microwave background and other sources, these results imply that the dark mattermotivations for Sommerfeld enhancements and GeV dark forces at present are, at best,strained.

We have discussed several possibly interesting directions for further study in Sec. V.Non-minimal models will generically include additional annihilation channels, such as thoseinvolving hidden Higgs bosons. These will generically strengthen the bounds on Seff. Wenoted also that the decay of the hidden sector Higgs boson to standard model particlescould be constrained by demanding consistency with the measured light element abundancesproduced during BBN. Additional features and constraints particular to specific models havebeen noted in Sec. V. We have focused on the 4µ final state here, which generically give thebest fits to the data. Different annihilation channels, such as 4e final states, are optimized atlower Seff and mX , but also yield worse fits to the data according to present estimates in theliterature. However, this conclusion may depend significantly on the cosmic ray propagationmodel parameters used. At present, no Sommerfeld models have been shown to explainthe PAMELA and Fermi excesses consistent with the constraint of thermal freeze out, andwe stress that complete and dedicated studies of freeze out are required to judge properlywhether more complicated models can self-consistently explain PAMELA and Fermi data.

The maximal Sommerfeld enhancements derived here, although not sufficient to explaincurrent PAMELA and Fermi excesses, are nevertheless significant enhancements. These willbe probed as PAMELA and Fermi continue to gather data, and, in the near future, at AMSon the International Space Station. The interpretation of future data as dark matter may, ofcourse, continue to be clouded by confusion from astrophysical uncertainties, but the diverseset of particle physics and astroparticle physics experiments that may probe weak-scale darkmatter in the near future at least provides some reason for optimism.

27

Acknowledgments

We thank Masahiro Ibe, Jason Kumar, Erich Poppitz, and Neil Weiner for useful conver-sations and correspondence, and Tracy Slatyer for motivating more detailed investigation ofresonant Sommerfeld enhancements and comments on an early draft of this work. We thankMarc Kamionkowski for pointing out the importance of inverse decays for the force carrierin equilibrium, and the participants of the Aspen Summer 2010 workshop “From Collidersto the Dark Sector” for stimulating discussions. The work of JLF and HY was supported inpart by NSF grants PHY–0653656 and PHY–0709742. The work of MK was supported inpart by NSF grant PHY–0855462 and NASA grant NNX09AD09G.

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